Developable surfaces are three-dimensional (3D) objects formed from two-dimensional (2D) surfaces that can be flattened to a plane without stretching the surface material; simple examples include the cylinder, the cone, and ruled surfaces. Developable surfaces are important in manufacturing because they can be created from pre-fabricated flat materials, such as paper, textiles, or sheet metal. They have been ubiquitous in engineering applications for centuries and there is an immense body of literature on them across the engineering sciences. The focus of the present work is on applications of developable surfaces for modeling primarily organic 3D shapes, taking perceptual factors related to assembly and visual appearance into account.
A number of existing techniques make it possible to approximate organic 3D shapes with developable surfaces. However, their scope is limited due to several factors, including tedious assembly and detracting visual artifacts in the assembled 3D object. On one hand, a 3D object made as a developable surface has to be sufficiently interesting and detailed to justify the effort and resources needed to assembly. On the other, the cost of the manufacturing generally increases with the complexity of the geometry.
Whereas the bulk of existing work addresses the creation of developable surface patches with little or no attention to assembly of physical models, U.S. Pat. No. 6,819,966 describes a general method for creating developable surface hardcopies from planar materials. The patent focuses how general cost functions can be used for automatic creation of triangulations for closed curve loops. Unfolding the triangulations into the plane yields developable strips that define the surface. It covers this particular problem in depth, but it does not discuss any details how the curves are created, how to avoid self-intersections in the unfolded triangle strips, and several of the other limitations discussed in the background section. The patent does list a number of assembly techniques based on protruding tabs that are either glued to corresponding (marked) areas on the matching surface with an adhesive or bent and tied together with wire. The use of protruding tabs is a general idea that has been used for decades in paper crafts, assembly of tin toys, and many other areas, but it often needs glue, tape, wires, or other foreign parts. The techniques described and depicted in U.S. Pat. No. 6,819,966 require too much tedious manual work to be practical for assembly of models of the geometric complexity that would be required to make the output models sufficiently appealing to be feasible to produce on a larger scale.
Other existing work focuses on the approximating the organic geometry with developable pieces subject to some error bounds. The organic 3D model is analyzed and divided into pieces that can be approximated by developable surface patches. The resulting developable primitives can be piecewise planar, such as triangular or quadrilateral mesh strips, or higher order surfaces such as cone splines or NURBS surfaces. Many such solutions have been proposed, but they suffer from the following shortcomings:                Limited control over the number of developable surfaces.        Limited control over the shape of the developable surfaces.        Limited control over the regularity of the developable surfaces.        Lack of attention to the ease-of-assembly of the developable surfaces.        Lack of attention to perceptual aspects of the aesthetics of the developable surfaces.        Most existing developable surface techniques aimed at organic shapes represent surfaces as infinitely thin manifolds, ignoring the fact that real-world materials have a non-zero thickness.        
In terms of the output surfaces, there is an important tradeoff:                Short developable surface strips are problematic because:                    They produce a large number of developable surfaces that complicate the assembly.            Short strips make it difficult or impossible to create stylized developable surfaces where seams appear as long, flowing curves.                        Long developable strips lead to two problems:                    Self-intersections occur after unfolding them into a plane, making it impossible to manufacture strips in one piece.            Irregular shapes may be difficult to pack optimally into a piece of planar material.                        Combinations of long, short, small, and large developable surfaces will not appear visually attractive unless careful attention is paid to the overall structure of the network of developable patches.        
In summary, a cohesive solution that addresses all these limitations remains an open problem. As a result, developable surfaces have seen limited use in the manufacturing of organic 3D geometry.